Question

How do you condense this expression into a single logarithm? $ \dfrac{{{{\log }_5}x}}{2} + \dfrac{{{{\log }_5}y}}{2} + \dfrac{{{{\log }_5}z}}{2} $

Answer 1

Hint: We have been given an expression containing three logarithmic terms. We have to use different properties of logarithm to simplify and write all the terms into one single term. For addition of logarithmic terms we have the property, $ {\log _a}x + {\log _a}y = {\log _a}\left( {xy} \right) $ .

Formula used:
 $
  {\log _a}x + {\log _a}y = {\log _a}\left( {xy} \right) \\
  {\log _x}\left( {{x^n}} \right) = n{\log _x}x = n \\
  \dfrac{{{{\log }_a}x}}{{{{\log }_a}y}} = {\log _y}x \;
  $

Complete step by step solution:
We have been given an expression $ \dfrac{{{{\log }_5}x}}{2} + \dfrac{{{{\log }_5}y}}{2} + \dfrac{{{{\log }_5}z}}{2} $ with three terms in logarithmic functions.
We have to condense this expression into a single logarithm, i.e. we have to simplify the given expression such that we are left with a single term of logarithmic function. We have to use properties of logarithmic functions for this purpose.
We can observe that all the three terms have the same denominator $ 2 $ , so we can write a common fraction with $ 2 $ as a denominator.
 $ \dfrac{{{{\log }_5}x}}{2} + \dfrac{{{{\log }_5}y}}{2} + \dfrac{{{{\log }_5}z}}{2} = \dfrac{{{{\log }_5}x + {{\log }_5}y + {{\log }_5}z}}{2} $
All the logarithmic terms have common base $ 5 $ .
We have a property for addition of logarithmic terms with common base given as,
 $ {\log _a}x + {\log _a}y = {\log _a}\left( {xy} \right) $
We can extend this property for three terms and write,
 $ {\log _a}x + {\log _a}y + {\log _a}z = {\log _a}\left( {xyz} \right) $
Here if we substitute $ a = 5 $ , we get,
 $ {\log _5}x + {\log _5}y + {\log _5}z = {\log _5}\left( {xyz} \right) $
We can use this in our expression to get,
 $ \dfrac{{{{\log }_5}x + {{\log }_5}y + {{\log }_5}z}}{2} = \dfrac{{{{\log }_5}\left( {xyz} \right)}}{2} $
Further we try to convert the denominator also in logarithmic function with base $ 5 $ .
We have a property of logarithm which states, $ {\log _x}\left( {{x^n}} \right) = n{\log _x}x = n $
So we can write $ 2 $ as,
 $ 2 = 2{\log _5}5 = {\log _5}\left( {{5^2}} \right) = {\log _5}25 $
Thus, our expression now becomes,
 \[\dfrac{{{{\log }_5}\left( {xyz} \right)}}{2} = \dfrac{{{{\log }_5}\left( {xyz} \right)}}{{{{\log }_5}25}}\]
For division of logarithm with common bases we have the property,
 $ \dfrac{{{{\log }_a}x}}{{{{\log }_a}y}} = {\log _y}x $
Therefore,
 \[\dfrac{{{{\log }_5}\left( {xyz} \right)}}{{{{\log }_5}25}} = {\log _{25}}\left( {xyz} \right)\]
Hence, the condensed form of the given expression is \[{\log _{25}}\left( {xyz} \right)\] .
So, the correct answer is “ \[{\log _{25}}\left( {xyz} \right)\] ”.

Note: We used different properties of logarithmic function to simplify the given expression into one term. While using properties we have to keep an eye on the base of the function, as most of the properties are for common bases. Also, we can expand the resulting term to arrive at the given expression using the same properties in reverse.